The example of the micro- to macroeconomic transition
is a really good one to get a sense of how coarse graining works.
So let's take an example here
to give you a sense of what we mean
when we say that we're going to coarse grain a system.
We're going to simplify it, we're going to renormalize it,
in the language of physics.
So, let's take just a partial description of the entire economy that we have today.
So, here's just a sample example.
James got a job; Mary was promoted; Sally was laid off;
Harold to parental leave; Xavier retired; Sunshine went back to graduate school.
Well, if you're a macroeconomist,
really all you care about is that three people lost their jobs,
or rather four people lost their jobs, and one person got a job,
and so the net change in employment was -3.
And so what the macroeconomist has done there,
in the construction of her variables,
is taken a complicated story and made it a lot simpler.
And in doing that, almost always the following thing happens,
which is that they confuse or mix up two entirely different descriptions.
So, if we imagine instead of James getting a job, it was Sunshine;
instead of Mary getting promoted, it was Sally;
and Mary got laid off.
Harold, in this case, didn't take parental leave.
Xavier kept working instead of retiring,
and James got sick and lost his job,
and Nolan went to graduate school.
These are very different descriptions at the micro-level;
but from the point of view
of the construction of the macro-level economic variables,
the number of people who were unemployed or employed
at a certain point in time,
those two descriptions are identical.
And so what coarse graining does in other words,
is merge states of the world.
We use a lot of different terms
over the course of this series of lectures.
One term is "equivalence classes".
So you can think these two descriptions
are in the same chronical equivalence class,
and that equivalence class contains all the descriptions of the world
in which the net change in employment was -3.
You can also think of these as if you're a computer scientist,
as irreversible mappings,
mappings that are are onto, but not necessarily, one-to-one.
If you're a physicist, you can even think of them, if you like –
and we'll talk about this a little bit
in the final part of this set of modules on renormalization –
if you're a physicist,
you can also think of these as irreversible transformations:
the entropy of the description, the uncertainty of the description,
has now gone down.
These two different micro-level descriptions
both map to the same final storing.
The macroeconomist – here's another way to put it –
the macroeconomist is much more certain about the world than the microeconomist.
Let's try to get a more intuitive sense of coarse graining,
and we'll give you a sense of how coarse graining plays out
in something that we understand very well,
which is the representation of images for the human mind.
This is a woodcut here from John Tenniel
of one of the episodes in "Alice in Wonderland".
Actually, it's from the second of the two books,
"Alice Through the Looking Glass".
You can see Alice here playing with a ball of yarn
and her kitten Dinah.
What I'd like you to imagine us doing here
is simplifying this image in different ways.
What we're going to do in particular is coarse grain the image
in the following fashion.
So, imagine dividing this image up into grids,
and then, at each grid point here –
if we were to zoom in to this digitized version of Tenniel's image,
if you're going to zoom in –
I'm going to zoom in just a little on Dinah's ear here –
if we were to zoom in, we'll see a little patch of that image.
And what we're going to say
is for each of these patches, we're going to represent them now.
Instead of all of the individual details – the square is white, the square is black –
instead of all the individual details,
we're going to have that patch just take a vote.
In fact, it's going to be a majority vote,
it's going to be like the electoral college of images here.
So that image, there it turns out as majority black pixels,
and so it's going to coarse grain to a single chunky black pixel.
And we're going to go grid square by grid square here,
taking all of the complexity of the image within that ten by ten chunk
and compressing it down to a single answer:
black or white, 1 or 0.
And if we go right next door on that image,
here's a different part of Dinah, different part of Alice's cat,
And what we can see here
is that even though this image patch at the fine grain level is different,
the two descriptions in the different parts of the image are different,
our course grading majority vote will represent them identically.
Another way to put it is:
if Vermont goes Democratic,
it's basically because everyone in Vermont is democratic.
If Pennsylvania goes Democratic –
there's a lot more non-Democratic votes in that state,
but because of how we're going to coarse grain the system,
we're going to say that Pennsylvania and Vermont
both like the Democratic candidate equally.
So, what happens when we do that?
Well, here's an example.
This is majority vote coarse graining.
What we can see here
is that on the left-hand side, we have the original image,
and on the right-hand side we have something
that kind of looks pretty much like what Alice sort of looks like,
but we're missing quite a lot.
Here's another case, this is a different rule.
In this case, here, instead of taking the majority vote,
I did what's called [decimation], at least in the literature,
and this is a term that was invented by the physicist Leo Kadanoff
when he came to do this kind of coarse graining,
not on pictures from "Alice in Wonderland",
but on images of atoms being spin up or spin down,
and we'll talk about that in a later module
when we come to the Ising model.
What Leo said to do is:
Don't even bother to average over that grid square,
just take the point in the top left corner,
and have that define it.
You can see in this case here, if we use that decimation rule,
that same patch of die now gets mapped, not to black, but to white.
So what I've introduced you to here is two ways to simplify an image,
two ways to coarse grain the image,
and, of course, different images will now coarse grain to the same thing.
I can make many microscopic changes in Tenniel's original illustration,
and still get back the same final image of Alice.
Of course, the microscopic changes that my image is insensitive to,
or that, rather, my coarse graining of the image is insensitive to,
the microscopic changes will depend upon
the coarse graining algorithm that I choose.
So if I do the majority vote,
then I'll be able to make some changes that won't show up,
and if I instead take the decimation coarse graining rule,
there's a different set of changes that I can make at the microscopic level
that will leave the system unchanged.